Just going by the spec sheet numbers, you've only got about 11 and a half percent of the trailer weight on the tongue. Now you put 800 pounds in the back of the trailer and you're going to have serious problems. You might consider moving everything you can to the front bedroom while you are Towing and putting one of those bikes in the bed of the pickup truck. When you've done everything you can, head to a cat scale. There are many threads that will teach you how to weigh your setup.
And no need to thank your dealer for his advice. A long bed F250 would have been a better match for what I believe will always be a difficult situation. We had a front load toy hauler that gave us the opposite problem but never did have a sway problem. And the dually really didn't give a hoot!